Problem: Max wrote an algorithm that searches for a specific term within a large set of terms. The following function gives the length of the search, in number of steps, over a set with $n$ terms: $S(n)=1.6\cdot\ln(0.9n)$ What is the instantaneous rate of change of the search length for a set of $10$ terms? Choose 1 answer: Choose 1 answer: (Choice A) A $0.16$ steps per term (Choice B) B $0.16$ steps per second (Choice C) C $3.5$ steps per term (Choice D) D $3.5$ steps per second
Explanation: Understanding the problem The function that represents the instantaneous rate of change of $S(n)$ is its derivative, $S'(n)$. Therefore, the instantaneous rate of change of the search length for a set of $10$ terms is $S'(10)$. Let's find $S'(n)$ and evaluate it at $n=10$. Finding $S'(n)$ $S'(n)=\dfrac{1.6}{n}$ Finding $S'(10)$ $\begin{aligned} S'(10)&=\dfrac{1.6}{10} \\\\ &= 0.16 \end{aligned}$ Interpreting units $S(n)$ is the number of ${\text{steps}}$ it takes for the search to go over $n$ ${\text{terms}}$. Therefore, we measure its rate of change in ${\text{steps}}$ per ${\text{term}}$. In conclusion, the instantaneous rate of change of the search length for a set of $10$ terms is $0.16$ steps per term. The rate of change is positive because the number of steps is increasing.